No Tight Projective Plane:

To show that there is no tight immersion of the
real projective plane,
Kuiper used the fact that a tight immersion of a surface M can
be decomposed into two components, M+ and M-, with
the following properties:

The curvature is non-negative on M+ and non-positive on
M-,

The image of M+ is an embedding and is equal to the
convex envelope
of the image of M minus a finite number of planar, convex
disks,

The boundary of each of these disks is the image of a curve in
M that doesn't bound a region in M.

The boundary curves mentioned in (3) are called
top cycles, and they
play a key role in understanding tight immersions. Note that a
top cycle is an embedded, planar, convex curve, and that every top
cycle has an
orientable neighborhood
.

If a tight immersion of a connected surface has no top cycles, then it is
necessarily a sphere, since in this case the image of the M+
region is all of the convex envelope (and there is no M-
region). So a tight immersion of the projective plane must have at
least one top cycle.

There is only one class of embedded curves on the projective plane
that do not bound regions; but curves in this class have
non-orientable
neighborhoods (the neighborhood is a
Möbius band, as in the
diagram below), and so they can not be top cycles.

This is a contradiction, so there can be
no tight immersion of the real projective plane.

There is no tight Klein bottleNon-orientable tight surfacesKuiper's initial question